A Conic Conjugate Gradient Method For Large-scale Optimization

Conic model is a generalization of quadratic model. Conjugate gradient method is one of the most effective methods for large-scale optimization. The combination of conic model and conjugate gradient techniques may form more powerful methods.The present conic conjugate gradient methods are processed generally by the following steps: at first we approximately change the conic model into the quadratic model, and then solve the quadratic model by many previous quadratic model algorithms, and finally get the optimal solution with the conic model. Maybe it could overlook the qualities of conic function in the process of approximative change. So we mainly study a new conic conjugate gradient algorithm and its modified algorithm without this change in the paper. The search direction of the new algorithm takes the horizon vector as a part of it and refers to the search direction of classic conjugate gradient methods. We prove the convergence of the new algorithm and carry out the numerical experiments.The paper is divided into five chapters. The first chapter briefly introduces the issues related to optimization and the contents of this paper. The second chapter describes the progress of study on conic conjugate gradient methods. In the third chapter we give the basis and structure of the search direction of the new algorithm, discuss the choice of the parameters in it and analyze its decent property. In the fourth chapter, by using the frame of classic conjugate gradient methods, we give a description of the new conic conjugate gradient algorithm for large-scale unconstrained optimization, and prove its convergence under some conditions. By adding a judgment criterion, we also give its modified algorithm. In the fifth chapter we carry out numerical comparison experiments about the algorithms in Chapter 4, analyze numerical results, and draw useful conclusions. The theoretic and numerical results show that the algorithm in this paper is efficient and promising.